About 10 blocks of the trendy neighborhood — including a community center and a portion of the High Line — began receiving free outdoor Wi-Fi yesterday courtesy of Google and the local Business Improvement District.

The global tech giant installed 29 antennae covering an area bounded roughly by Eighth to 10th avenues and West 15th to West 19th streets, but including sections as far west as the West Side Highway and as far south as Gansevoort Street. Google employs 3,000 people at its building within the zone.

The network instantly became the largest contiguous free Wi-Fi zone in the city, which currently offers Wi-Fi in libraries and at 20 parks.

Sen. Chuck Schumer pronounced the achievement “great, great” and marveled that the installation cost was a modest $250,000. In fact, officials later said the price tag was closer to $115,000, plus $45,000 a year for maintenance.

“It’s not very expensive,” Schumer said at a press conference. “The mayor and I are talking. Maybe we can do this for the whole city.”

“Federal money?” interjected Mayor Bloomberg. “We’d love to do it.”

Dan Biederman, president of the Chelsea Improvement Co. business group, said Google quickly agreed to partner on the Wi-Fi project, which took about 18 months to launch.

Biederman also runs the Bryant Park BID, the first park in the city to offer free Internet connections, and the 34th Street BID.

He said that within a year he hopes to have free Wi-Fi in the Herald Square area.

But the signal in Chelsea won’t be available indoors, except for the local community center at a nearby Housing Authority complex.

New York, which is trying to position itself as the digital capital of the nation, lags many other cities.

“I’ve been beating my head against the wall,” said City Councilwoman Gale Brewer (D-Manhattan), former chair of the Technology Committee, referring to her decade-long effort to expand free Wi-Fi.

Rachel Haot, the city’s chief digital officer, said it would be “enormously complex and expensive” to hot-spot all of New York because of its density, geography and large number of tall structures.

Bloomberg said the issue comes down to money. “Somebody’s got to pay,” he said.